A new meshless method using Taylor series to solve elasticity problems

Authors

  • Yendoubouam Tampango Laboratoire LEM3, UMR CNRS 7239, Ile du Saulcy, 57045 Metz cedex 01, France
  • Michel Potier-Ferry Laboratoire LEM3, UMR CNRS 7239, Ile du Saulcy, 57045 Metz cedex 01, France
  • Yao Koutsawa CRP Henri Tudor Luxembourg, 66 Rue de Luxembourg, L-4221 Esch sur Alzette, Luxembourg
  • Salim Belouettar CRP Henri Tudor Luxembourg, 66 Rue de Luxembourg, L-4221 Esch sur Alzette, Luxembourg

DOI:

https://doi.org/10.13052/17797179.2012.721500

Keywords:

PDE, meshless method, convergence analysis, Taylor series expansion, Domb– Sykes plot

Abstract

A meshless method is presented and analysed. In this approach, one discretises only the boundary, the partial differential equation being solved in the domain by using Taylor series expansion. A least square method is used to apply boundary conditions. In this paper, the method is applied to Navier equations for linear elasticity. Various tests are presented to discuss the efficiency and robustness of the method. The convergence is exponential with respect to the degree but it depends on the radius of convergence of the series. That is why an algorithm has been associated with the Domb–Sykes plot that is a classical method to detect singularities and evaluate the radius of convergence.

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References

Babuska, I., Szabo, B.A., & Katz, I.N. (1981). The p-version of the finite element method. SIAM Journal

on Numerical Analysis, 18, 515–545.

Baker, G.A., & Graves-Morris, P. (1996). Padé approximants. Cambridge: Cambridge University Press.

Campion, S.D., & Jarvis, J.L. (1996). An investigation of the implementation of the p-version finite element

method. Finite Elements in Analysis and Design, 23(1), 1–21.

Domb, C., & Sykes, M.F. (1961). Use of series expansions for the Ising model susceptibility and

excluded volume problem. Journal of Mathematical Physics, 2(1), 63–67.

Garajeu, D., Cochelin, B., & Medale, M. (2010). Analyse et amélioration des séries (Analysis and

improvement of series), Technical report, Laboratoire de Mécanique et d’ Acoustique.

Hunter, C., & Guerrieri, B. (1980). Deducing the properties of singularities of functions from their Taylor

series coefficients. SIAM Journal on Applied Mathematics, 39(2), 248–263.

Muskhelishvili, M. (1958). Some basic problems of the mathematical theory of elasticity (pp. 113–115).

Groningenp: Noordhoff.

Tampango, Y. (2009). Résolution des problèmes incompressibles de Stockes par la méthode de perturbation

(Solving incompressible Stokes problems by a perturbation method) (Master’s thesis, Université

de Metz).

Tampango, Y., Potier-Ferry, M., Koutsawa, Y., & Belouettar, S. (2012). Convergence analysis and detection

of singularities within a boundary meshless method based on Taylor series. Engineering Analysis with

Boundary Elements, 36(10), 1465–1472.

Taro, M. (2012). Elaboration d’ un algorithme numérique permettant de résoudre un problème d’ élasticité

non linéaire à l’ aide d’ une méthode developpée au LEM3 (A numerical algorithm to solve a nonlinear

elasticity problem by a method developed in LEM3)(Master’s thesis, Université de Lorraine).

Van Dyke, M. (1974). Analysis and improvement of perturbation series. Quarterly Journal of Mechanics

and Applied Mathematics, 27, 423–450.

Zézé, D. (2009). Calcul de fonctions de forme de haut degré par une technique de perturbation (Computating

high degree shape functions by a perturbation technique) (PhD thesis, Université Paul Verlaine

Metz).

Zézé, D.S., Potier-Ferry, M., & Damil, N. (2010). A boundary meshless method with shape functions

computed from the PDE. Engineering Analysis with Boundary Elements, 34(8), 747–754.

Zhang, X., Liu, X.H., Song, K.Z., & Lu, M.W. (2001). Least-squares collocation meshless method.

International Journal for Numerical Methods in Engineering, 51(9), 1089–1100.

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Published

2012-06-06

How to Cite

Tampango, Y., Potier-Ferry, M. ., Koutsawa, Y. ., & Belouettar, S. . (2012). A new meshless method using Taylor series to solve elasticity problems. European Journal of Computational Mechanics, 21(3-6), 365–373. https://doi.org/10.13052/17797179.2012.721500

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